Explainable Machine Learning for Multilayer Planar Winding Inductance Estimation

Spyros Rigas; Theofilos Papadopoulos; Georgios Alexandridis; Antonios Antonopoulos

cs.LGSep 29, 2026

Explainable Machine Learning for Multilayer Planar Winding Inductance Estimation

Authors: Spyros Rigas, Theofilos Papadopoulos, Georgios Alexandridis, Antonios Antonopoulos

Organizations: Department of Digital Industry Technologies, National and Kapodistrian University of Athens, Greece · School of Electrical and Computer Engineering, National Technical University of Athens, Greece

Abstract

Rapid and accurate self-inductance estimation for multilayer rectangle-shaped planar windings is essential for modern high-frequency power converters, yet traditional workflows rely on complex mathematical equations, rigid monomial formulas or unexplainable black-box machine learning (ML) models that degrade severely outside their training domain. This paper introduces an explainable ML framework unifying post-hoc feature attribution (SHAP and permutation importance) with Kolmogorov-Arnold Network-guided symbolic regression via the SR-KAN framework to discover closed-form analytical equations without prior structural assumptions. Evaluated on a new open-source dataset of over 10,000 Finite Element Analysis (FEA) simulations across seven out-of-distribution (OOD) classes, standard tree-based ensembles exhibit severe extrapolation errors (> 36%), whereas the unconstrained SR-KAN expression achieves a robust OOD relative error of 8.22%. Experimental verification across 55 physical printed circuit board prototypes (up to 8 layers, with inductances from 4.11 μH to 559.27 μH) confirms that the KAN-discovered expression translates effectively to real-world hardware, predicting inductance with a mean absolute relative error of 6.26%. To support reproducible research, the complete FEA simulation dataset and prototype measurements are released open-source.

Figures & tables

Panel of Figure 1 — Fig. 1: Geometric parameters for a typical MLRPW (4-layered).

Parameter	Value Range		# of Values	Comments
$D_{1}$	70:10:160 1	mm	10	$D_{1}\leq D_{2}$
$D_{2}$	70:10:160 1	mm	10	$D_{1}\leq D_{2}$
$d_{1}$	$\in[17,123]$	mm		derived from ( 1 )
$d_{2}$	$\in[17,123]$	mm		derived from ( 1 )
$w$	3, 4, 5	mm	3
$s$	0.1, 0.3, 0.5	mm	3

TABLE I: Value Ranges for Dataset Parameters

	Core Split		OOD Split
Model	$\mathbf{R^{2}}$ (%)	$\boldsymbol{\mathcal{E}}$ (%)	$\mathbf{R^{2}}$ (%)	$\boldsymbol{\mathcal{E}}$ (%)
RF	$99.90$	$1.87\pm 0.02$	$52.53$	$36.65\pm 0.00$
XGBoost	$99.94$	$1.43\pm 0.02$	$51.29$	$37.45\pm 0.00$
SVR	$99.66$	$3.08\pm 0.05$	$94.18$	$12.95\pm 0.00$
Reference MLR [ 23 ]	$98.71$	$5.88\pm 0.17$	$98.10$	$9.92\pm 0.00$

TABLE II: Predictive Accuracy Comparison of Machine Learning Models Within the Core and OOD Splits

Feature	RF	XGBoost	SVR	RF	XGBoost	SVR
	Permutation Importance			SHAP
$D_{1}$	$0.0150$	$0.0087$	$0.1030$	$0.0769$	$0.0482$	$0.1350$
$D_{2}$	$0.0004$	$0.0028$	$0.0071$	$0.0044$	$0.0258$	$0.0333$
$\bar{D}_{1}$	$0.0823$	$0.0609$	$0.3152$	$0.1135$	$0.1326$	$0.2459$
$\bar{D}_{2}$	$0.0482$	$0.0234$	$0.0109$	$0.0881$	$0.0679$	$0.0407$
$w$	$0.0164$	$0.0210$	$0.0027$	$0.0483$	$0.0733$	$0.0220$

TABLE III: Normalized Feature Importance Scores Across Machine Learning Models

Fig. 2: Consensus Ranking (

CR

) scores across the 9-dimensional MLRPW geometric space, showing the stacked contribution of mean permutation importance and mean SHAP attribution.

Fig. 3: Comparison plots mapping the predicted self-inductance

L_{\text{pred}}

against the true self-inductance

L_{\text{true}}

for the original monomial (left column) and the refined MLR model (right column) evaluated across both the core split (top row) and the OOD split (bottom row). The dashed diagonal represents the ideal zero-error line (

L_{\text{pred}}=L_{\text{true}}

	Core Split		OOD Split
Model	$\mathbf{R^{2}}$ (%)	$\boldsymbol{\mathcal{E}}$ (%)	$\mathbf{R^{2}}$ (%)	$\boldsymbol{\mathcal{E}}$ (%)
Refined MLR	$99.51$	$4.12\pm 0.03$	$98.98$	$7.91\pm 0.00$
PySR	$99.29$	$4.74\pm 0.17$	$96.14$	$17.32\pm 3.17$
SR-KAN	$99.49$	$3.77\pm 0.04$	$98.42$	$8.22\pm 0.55$

TABLE IV: Predictive Accuracy Comparison of Symbolic Regression Frameworks Within the Core and OOD Splits

Fig. 4: Relative residuals

(L_{\text{true}}-L_{\text{pred}})/L_{\text{true}}

plotted against the target self-inductance

L_{\text{true}}

on the OOD split for the refined MLR, PySR and SR-KAN models. The individual error points are color-coded according to the seven distinct OOD classes.

Physical Parameter	Minimum	Maximum	Unit
Outer dimension $D_{1}$	70.0	210.0	mm
Outer dimension $D_{2}$	100.0	294.0	mm
Trace width $w$	3.0	7.0	mm
Trace spacing $s$	0.1	2.0	mm
Turns per layer $N_{T}$	4	12	turns
Layer count $N_{L}$	1	8	layers

TABLE V: Experimental Prototype Ranges and FEA Validation Summary (56 Samples)

Fig. 5: Comparison plots mapping the measured self-inductance

L_{\text{meas}}

against the estimated self-inductance

L_{\text{equations}}

for equations ( 5 ), ( 7 ), and ( 8 ).

Explore similar work

Comparative Evaluation of Machine Learning and Deep Learning Models for Wound-Rotor Synchronous Motor Performance Prediction
Jun 19, 2026Kıvanç Doğan, Ahmet OrhanNeural Surrogate Modeling
Fırat University, Faculty of Engineering, Department of Electrical and Electronics Engineering, Elazığ
AttentionCap: Transformer Based Capacitance Matrix Learning Toward Full-Chip Extraction
Jun 6, 2026Jiechen Huang, Hector R. Rodriguez, Dingcheng Yang +3Electronic Design Automation Transformer Attention
Dept. Computer Science & Tech., BNRist, Tsinghua Univ., Beijing, China · School of IC, BNRist, Tsinghua Univ., Beijing, China · School of IC, Peking Univ., Beijing, China
Using Large Language Models and Knowledge Graphs to Improve the Interpretability of Machine Learning Models in Manufacturing
Apr 17, 2026Thomas Bayer, Alexander Lohr, Sarah Weiß +2Manufacturing LLM Grounding
University of Applied Sciences Ravensburg-Weingarten, Germany

Jun 19, 2026J/K move · Enter open · S save

Comparative Evaluation of Machine Learning and Deep Learning Models for Wound-Rotor Synchronous Motor Performance Prediction

Kıvanç Doğan, Ahmet Orhan

Fırat University, Faculty of Engineering, Department of Electrical and Electronics Engineering, Elazığ

Neural Surrogate Modeling

Open paper AskPDF arXiv

Wound rotor synchronous motors have emerged as a strong alternative that eliminates dependence on REEs. However, WRSM design requires the simultaneous optimization of numerous geometric and electromagnetic parameters, and the high computational cost of conventional finite element analysis severely limits the rapid exploration of the large parameter space. Although there are machine-learning-based surrogate modeling studies in the literature, they generally compare only a limited number of models, exclude deep learning architectures, and do not provide a comprehensive benchmark specific to WRSM. In this study, the performance of a total of eight machine learning and deep learning models from four different algorithmic families was systematically compared for the prediction of WRSM torque and motor efficiency. On a dataset of 3351 samples generated using Latin Hypercube Sampling in the Motor-CAD simulation environment, each model was trained with 10 different random seed values and tuned via Optuna hyperparameter optimization. Different from the existing literature, this study jointly offers a broad model spectrum including recent deep learning architectures such as FT Transformer, a multi-seed reproducibility protocol, and a Pareto analysis of the computational cost-accuracy trade-off. The results revealed that neural-network-based models systematically outperform tree-based models. The FT-Transformer model achieved the highest single-model accuracy with R^2 = 0.9928, producing predictions in 0.33 milliseconds and thus obtaining several orders of magnitude speedup compared to FEA. Model performances were evaluated in a multidimensional manner using R^2, MAE, RMSE, and MAPE metrics.

Abstract

Explore similar work

Comparative Evaluation of Machine Learning and Deep Learning Models for Wound-Rotor Synchronous Motor Performance Prediction

AttentionCap: Transformer Based Capacitance Matrix Learning Toward Full-Chip Extraction

Using Large Language Models and Knowledge Graphs to Improve the Interpretability of Machine Learning Models in Manufacturing